Autonomous Spectrum Balancing for Digital Subscriber Lines

Home , DSL modem, Modem, Node (networking)

IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 55, NO. 8, AUGUST 2007 4241 Autonomous Spectrum Balancing for Digital Subscriber Lines Raphael Cendrillon, Member, IEEE, Jianwei Huang, Member, IEEE, Mung Chiang, Member, IEEE, and Marc Moonen, Fellow, IEEE

Abstract—The main performance bottleneck of modern digital formance improvement in modern DSL systems remains to be subscriber line (DSL) networks is the crosstalk among different crosstalk, which is the interference generated among different lines (i.e., users). By deploying dynamic spectrum management lines in the same cable binder. The crosstalk is typically 10–20 (DSM) techniques and reducing excess crosstalk among users, a network operator can dramatically increase the data rates and ser- dB larger than the background noise, and direct crosstalk cance- vice reach of broadband access. However, current DSM algorithms lation (e.g., [1], [2]) is difficult in many cases, due to complexity suffer from either substantial suboptimality in typical deployment (both amount of computation needed and the requirements for scenarios or prohibitively high complexity due to centralized com- new chip sets) or unbundling requirement (i.e., incumbent ser- putation. This paper develops, analyzes, and simulates a new suite 1 of DSM algorithms for DSL interference-channel models called au- vice providers must rent certain lines to their competitors). tonomous spectrum balancing (ASB). The ASB algorithms utilize Recently, various dynamic spectrum management (DSM)2 al- the concept of a “reference line,” which mimics a typical victim line gorithms have been proposed to address this frequency-selec- in the interference channel. In ASB, each modem tries to minimize tive interference problem by dynamically optimizing transmis- the harm it causes to the reference line under the constraint of sion power spectra of different modems in DSL networks. DSM achieving its own target data-rate. Since the reference line is based on the statistics of the entire network, rather than any specific algorithms can significantly improve data rates over the cur- knowledge of the binder a modem operates in, ASB can be imple- rent practice of static spectrum management, which mandates mented autonomously without the need for a centralized spectrum spectrum mask or flat power backoff across all frequencies (i.e., management center. ASB has a low complexity and simulations tones). using a realistic simulator show that it achieves large performance gains over existing autonomous algorithms, coming close to the This paper develops, analyzes, and simulates a suite of DSM optimal rate region in some typical scenarios. Sufficient conditions algorithms for power allocation (or, equivalently, bit loading), for convergence of ASB are also proved. called autonomous spectrum balancing (ASB). Overcoming the Index Terms—Digital subscriber lines (DSLs), distributed al- bottlenecks in the state-of-the-art DSM algorithms, ASB is a set gorithm, dual decomposition, interference channel, multicarrier, of algorithms that, simultaneously, is autonomous (distributed power allocation, spectrum management. algorithm across the users without explicit real-time information exchange), has low complexity, is provably convergent under certain sufficient conditions, and achieves rate region I. INTRODUCTION close to the global optimum. The methods of “static pricing” A. Motivation and “frequency-selective waterfilling” developed in ASB may also be of interest to the general problems of decoupling cou- IGITAL SUBSCRIBER LINE (DSL) technologies trans- pled objective function and of multicarrier interference channel. Dform traditional voice-band copper channels into broadband access systems, which are typically capable of delivering B. Related Work on DSM Algorithms data rates of several Mb/s per twisted-pair over a distance of 10 One of the first DSM algorithms is the Iterative Water-filling kft in the basic asymmetric DSL (ADSL). Despite over 160 mil- (IW) algorithm [3], where each line maximizes its own data rate lion DSL lines worldwide as of 2006, the major obstacle for per- by waterfilling over the noise and interference from other lines. The IW algorithm is autonomous, has a linear complexity in the Manuscript received September 1, 2006; revised November 30, 2006. The number of users and number of frequency tones, and has been associate editor coordinating the review of this manuscript and approving it for shown to converge in typical DSL deployments, e.g., [3], [4]. publication was Prof. Brian L. Evans. This work was supported in part by Al- catel-Bell and by the US NSF Grant CNS-0427677 and CCF-0448012. A por- Although IW can achieve near optimal performance in weak in- tion of this paper was has appeared in the IEEE Proceedings of the International terference channels, it is highly-suboptimal in the widely-en- Symposium of Information Theory, July 2006. countered near–far scenarios (which will be described in detail R. Cendrillon is with Marvell Hong Kong, Ltd., Mongkok, Hong Kong (e-mail: [email protected]; [email protected]). in Section II), such as mixed central office and remote terminal J. Huang and M. Chiang are with Department of Electrical Engi- neering, Princeton University, Princeton, NJ 08544 USA (e-mail: jianweih@ 1Although in an unbundled network DSM can be applied to in-domain lines, princeton.edu; [email protected]). in many cases out-of-domain lines cannot be coordinated, leading to some sub- M. Moonen is with the Department of Electrical Engineering, Katholieke optimality. Similarly the network management center can be used to coordinate Universiteit Leuven (K.U. Leuven,), ESAT/SISTA, B-3001 Leuven-Heverlee, lines in a centralized fashion, however such a network management center would Belgium (e-mail: [email protected]; [email protected]. require full knowledge of the network topology, which is often difficult to im- be). plement in practice. Further discussion can be found in Section I-B. Color versions of one or more of the figures in this paper are available online 2The DSM algorithms discussed in this paper are different from the “dynamic at http://ieeexplore.ieee.org. spectrum sharing” algorithms, which are used to refer to opportunistic sharing Digital Object Identifier 10.1109/TSP.2007.895989 of the spectrum resources in wireless communications.

TABLE I COMPARISON OF VARIOUS DSM ALGORITHMS

deployments of ADSL and upstream VDSL. This is in part due While the band preference method calculates the bitloading in to the greedy and selfish nature of the algorithm. a distributed fashion, the band-preference coefficients (which Recently two optimal but centralized DSM algorithms were correspond to a spectral mask imposed on the waterfilling al- proposed, the Optimal Spectrum Balancing (OSB) algorithm gorithm) need to be calculated in some way, centralized or dis- [5] and the Iterative Spectrum Balancing (ISB) algorithm [6], tributed. This often requires the use of a centralized spectrum [7]. The OSB algorithm addresses the spectrum management management center. The performance of the band preference problem through the maximization of a weighted rate-sum method depends on the choice of the specific spectrum man- across all users, which explicitly takes into account the damage agement algorithm used. done to the other lines when optimizing each line’s spectra. IW, OSB, ISB, and SCALE mentioned above all assume syn- OSB has an exponential complexity in the number of users, chronous transmissions of the modems, which allows crosstalk making it intractable for DSL network with more than five to be modeled independently on each tone. This synchroniza- lines. As an improvement over the OSB algorithm, ISB was tion is rarely true in practice. Instead, the signal transmitted on proposed to implement the weighted-rate sum optimization in a particular tone of one modem will appear as crosstalk on a an iterative fashion over the users. This leads to a quadratic broad range of tones on the other modems. This inter-carrier- complexity in the number of users, which makes the ISB interference (ICI) complicates the DSM problem further. The feasible for networks with a relatively large number of users. state-of-the-art results for asynchronous transmissions are the However, an even more critical issue is that both OSB and two centralized greedy algorithms proposed in [10], bit-sub- ISB are centralized algorithms, which rely on a centralized net- tracting and bit-adding algorithms. Both algorithms start from work management center (NMC) to optimize the power spectral the power spectral density (PSD) obtained with the ISB algo- density (PSD) for all modems. The NMC requires knowledge rithm in the synchronous case, and search for local optimal solu- of the crosstalk channels among all lines and all background tions in the neighborhood by taking ICI into account. But again noise. Identification and transmission of crosstalk channel mea- these are centralized algorithms. surements back to the NMC are not supported in existing standards either. The operation of NMC requires a lot of overhead, C. Summary of Contributions in terms of both bandwidth and infrastructure. Furthermore, reg- The advantages of ASB algorithms are summarized as fol- ulatory requirements on unbundling service make it impossible lows. First of all, ASB is autonomous: it can be applied in a to perform a centralized optimization. Finally, many lines in the distributed fashion across users with no explicitly information same binder terminate on different quad cards in the DSL Ac- exchange in real-time. Furthermore, the algorithm has low com- cess Multiplexer because customers in the same neighborhood plexity in both the number of users and tones, and is proved to sign up for service at different times, which makes it hard to be convergent under certain sufficient conditions on the channel have central coordination even if one can tolerate the costs. gains. In the synchronous case, the ASB algorithm has a sim- A semi-centralized DSM algorithm called SCALE is pro- ilar complexity as IW, but in the near–far scenario achieves a posed in [8]. SCALE algorithm achieves better performance performance much better than IW and very close to ISB and than IW with comparable complexity. However, the algorithm OSB. In the asynchronous case, the ASB algorithm reduces the is not autonomous since explicit message passing among users complexity from those in [10], and achieves significant better is required. Such explicit real-time message passing in an un- performance than the ASB algorithm that does not consider the coordinated fashion requires modems to have sophisticated ICI. These features are obtained despite the convexity and cou- processing capabilities not available in DSL modems, including pling in the optimization problem of DSM. The comparisons be- blind synchronization, blind identification of the crosstalk tween ASB algorithms and other existing algorithms are listed channel, blind detection of the transmit constellation used by in Table I. It compares various aspects of different DSM algo- the crosstalk, and blind detection of the crosstalk signal. rithms, where ASB attains the best tradeoff among distributive- The band preference method is a practical way of imple- ness, complexity, and performance. Here we use to denote menting an optimized DSM PSD in a distributed fashion [9]. the number of tones and to denote the number of users. CENDRILLON et al.: AUTONOMOUS SPECTRUM BALANCING 4243

copper lines in the network. The first line is from the central office (CO) to customer 1. Since customer 2 is far away from CO, the service provider deploys a remote terminal (RT) near the edge of the network, which connects with customer 2 through a relatively short copper line. In the downstream transmission case shown in the figure, the transmitting modems (TX) are located at the CO and RT, and the receivers (RX) are at the customer homes. Each DSL modem transmits over Fig. 1. Mixed CO/RT distribution. multiple frequency tones (carriers). Multiple lines sharing the same binder generate crosstalk (interference) to each other on all frequency tones. Although the RT extends the footprint of The key idea behind ASB is to leverage the fact that DSL the DSL network, it also generates excessive interference to interference channel gains are very slowly time-varying, which the CO line due to the physical proximity between the RT TX enables an effective use of the concept of “reference line” that and the CO RX and since the two lines are in the same binder. represents a typical victim line. Roughly speaking, the reference However, CO TX generates little crosstalk to RT RX due to the line represents the statistical average of all victims within a typ- long distance between them. ical network, which can be thought as a “static pricing”. This A similar near–far problem also occurs in the upstream trans- differentiates the ASB algorithm with power control algorithms mission for VDSL, which operates at a frequency band up to in the wireless setting, where pricing mechanisms have to be 12 MHz, and line lengths are typically limited to less than 1.2 adaptive to the change of channel fading states and network km [12], [13]. As a result, VDSL modems are typically de- topology, or Internet congestion control, where time-varying ployed at one point in the network (e.g., a RT node), thus do not congestion pricing signals are used to align selfish interests for have the mixed CO/RT problem in the downstream transmis- social welfare maximization. By using static pricing, no explicit sions. However, due to the difference in customer home loca- message passing among the users is needed and the algorithm tions, shorter lines exhibit strong crosstalks into the longer lines becomes autonomous across the users. When adapting its PSD, receivers in the upstream transmissions. Furthermore, in mixed each line attempts to achieve its own target rate while mini- VDSL/ADSL deployments, RT-deployed VDSL will interfere mizing the damage it does to the reference line. We show such with the CO-deployed ADSL signals in the downstream. mechanisms can attain the balance between selfish and socially Next we introduce the mathematical models for both the syn- responsible operation. At the same time, each user in ASB still chronous and asynchronous transmission cases, following the keeps a local “dynamic pricing” of the individual power con- notation in [5], [6], and [10]. straint, which enables its own optimization problem to be de- A. Synchronous Transmission coupled across the tones within each user. We prove the convergence of ASB under an arbitrary number of users, for both Consider a DSL network with a set of users sequential and parallel updates. Since IW can be recovered as (i.e., lines, transmitting modems) and tones a special case of ASB in the synchronous case, our proof tech- (i.e., frequency carriers). Assuming the standard synchronous niques extend those in previous work on IW [3], [11]. discrete multi-tone (DMT) modulation, transmissions can be The rest of the paper is organized as follows. We intro- modeled independently on each tone as follows: duce the system model in Section II, for both synchronous and asynchronous transmission cases. The spectrum management problem and a general framework of ASB are outlined The vector contains transmitted signals in Section III. ASB algorithms for the synchronous and on tone , where is the signal transmitted by user at tone asynchronous cases will be given in Sections IV and V, respec- . Vectors and have similar structures: is the vector tively. We provide convergence proofs and simulation results of received signals on tone ; is the vector of additive noise in Sections VI and VII. The complexity properties of the ASB on tone and contains thermal noise, alien crosstalk and radio algorithm and the IW algorithm are given in Section VIII, and frequency interference. We denote the channel gain from trans- we conclude in Section IX. mitter to receiver on tone as . We denote the transmit Power Spectral Density (PSD) , where II. SYSTEM MODEL denotes expected value, and denotes inter-carrier spacing. ASB can be applied to many network topologies. In this The vector containing the PSD of user on all tones as paper we will often examine a typical near–far deployment for . downstream ADSL transmissions with a frequency band up to When the number of interfering users is large, the interference 1.1 MHz,3 as shown in Fig. 1, since it is one of the scenarios can be well approximated by a Gaussian distributed random where DSM techniques can give significant performance im- variable. The achievable bit rate of user on tone is defined provement. In this scenario there are at least two twisted-pair as

3The near–far problem does not occur in the upstream ADSL case, where the transmission frequency band is below 138 kHz and crosstalk is minimal at such (1) low frequencies. 4244 IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 55, NO. 8, AUGUST 2007 where is the normalized crosstalk Here denotes the target rate of user , and we can channel gain from user to user , and pick an arbitrary user to be user 1. Due to interference between is the normalized noise power density. Here de- users, Problem (3) is nonconvex. Furthermore, it is highly cou- notes the SINR-gap to capacity, which is a function of the de- pled across users (due to crosstalk) and tones (due to total power sired BER, coding gain and noise margin [14]. For notational constraint as well as ICI in the asynchronous case), making it a simplicity, we absorb into the definition of and . The very difficult optimization problem to solve. bandwidth of each tone is normalized to 1. Each user is typ- The rate region achieved by all users is convex in the asymp- ically subject to a total power constraint , due to the limita- totic case when number of tones becomes large [5]. Thus by tions on each modem’s analog frontend: . The changing the values of of all users , the solu- data rate on line is thus . tions of Problem (3) can trace out the Pareto optimal boundary of the rate region. B. Asynchronous Transmission It appears that any algorithm that globally solves (3) must have knowledge of all crosstalk channels and background noise In practice, it is often difficult to maintain perfect syn- spectra, forcing it to operate in a centralized fashion. In order to chronization between different DMT blocks due to different overcome this difficulty, we observe that for optimal solutions transmission delays on different lines. Compared with the of (3), each user adopts a PSD that achieves a fair compromise synchronous transmission case, here the received PSD of user between maximizing their own data-rate and minimizing the on tone , , also depends on other users’ transmit damage they do to other users. Based on this insight, we intro- PSD on tones other than tone , duce the concept of reference line, a virtual line that represents a typical victim user within the DSL system. One choice, but not the only one, for the reference line is to set it as the longest line seen within a network, which tends to have the weakest direct channel and see the worst crosstalk spectrum. Then, instead of solving (3), each user tries to maximize the achievable rate on the reference line, subject to its own rate and total power constraints. Here is the ICI coefficients estimated in the worst case [10], Note that the reference line is a fictitious line, and is used to represent a typical victim in a DSL network. This is inde- pendent of a particular binder, as no specific knowledge of a binder’s configuration is assumed. As such no centralized control is necessary, and the algorithm can be implemented in an autonomous fashion. The only knowledge a modem needs is its and has the symmetric and circular properties, i.e., direct channel, background noise and the distance from the CO . Then the achievable bit rate of user on tone to the RT if it is RT distributed. All of this information can ei- in (1) needs to be revised as (with set to 1) ther be measured locally, or, in the case of the CO to RT distance, can be programmed at the time that the RT is installed. This allows ASB to be implemented in an autonomous fashion during run-time, with the PSD and bitloading calculated locally. Since the main purpose of introducing the reference line is to characterize the damage that each user does to other interfering (2) users, we will make the achievable rate of the reference line user-dependent. In other words, from user ’s point of view, the where . All the other system parame- reference line’s rate is , where the achievable ters and constraints are the same as the synchronous case.4 bit rate on tone in the synchronous case is defined as

III. SPECTRUM MANAGEMENT PROBLEM AND THE GENERAL (4) FRAMEWORK OF ASB We consider the following spectrum management problem and, in the asynchronous case, as

(3) (5)

4While windowing [15] at the transmitter and receiver can be used to lower the DMT sidelobes and help reject ICI, in our experience a high level of ICI still remains, leading to significant performance degradation. Thus it is an important Intuitively, the reference line serves as a penalty term in each problem to mitigate ICI through DSM techniques. user’s optimization problem to align selfish behavior with social CENDRILLON et al.: AUTONOMOUS SPECTRUM BALANCING 4245 welfare maximization, and eliminates the need of explicit mes- on extensive field measurements and give a good representa- sage passing among users. Thus, instead of solving Problem (3) tion of the typical crosstalk channels seen in practice. Alter- which requires global information, we let each user solve the natively, it is also possible for the operator to use their own following problem in ASB algorithm: crosstalk channel models based on measurements made within their specific network, or with more advanced channel models which take into account both the inter-pair distance and non- ideal twisting of the twisted pairs within a binder [17]. For the (OPT1) empirical models used in standards, the only information needed to calculate the crosstalk channel is the length of the reference line, and the distance from the CO to the RT if a modem is RT distributed. All this information can be pre-set by the net- We want to emphasize that the each user autonomously solves work operator at the time that a modem is installed. Although a different version of Problem (OPT1). For user , Problem it may be possible to update this information periodically over (OPT1) only involves optimization over its own PSD , which the timescales of months or longer, such procedures are not re- determines the achieved rates of itself and the reference quired for the operation of the ASB algorithm. line . The interference generated by other users are con- Remark 2: The ASB algorithms use a static background noise sidered as fixed background noise in the optimization, and the spectrum for the reference line , which is set to the line noise achieved rates of other users in the network do not need to be seen by the reference line in the absence of self-crosstalk, i.e., considered. After each user solves its own version of Problem crosstalk from other DSL systems. In our experience, using this (OPT1), the crosstalk values change accordingly. Then each choice for the reference noise leads to good performance in user has to solve Problem (OPT1) again, repeating the process a broad range of scenarios. We believe the reason for this is until the PSD converges. The complete ASB algorithms will be that in most typical DSL deployments, the shorter lines, which given the Sections IV and V, where each version of ASB de- could potentially cause severe crosstalk to the weaker lines in ploys a unique way of solving Problem (OPT1). In Section VII, the system, will be configured such that they reduce their PSDs we will use “area of the rate region” as the performance metric in the frequencies where the weaker lines are active. As a result, when comparing ASB algorithms with other existing DSM al- in a DSL deployment with a reasonable distribution of rates, gorithms (e.g., [3], [5]–[7], [10]). each line should expect to see only a marginal increase in its To facilitate the analysis in the following sections, we also background noise spectrum due to crosstalk from the other lines consider a variation of Problem (OPT1), where we relax user in the system. This provides an intuitive explanation why the ’s target rate constraint and replace the optimization objective choice of self-crosstalk-free reference noise yields good perfor- by a weighted rate sum of user ’s own rate and the reference mance. Mathematically, it means that the specific engineering line’s rate seen by user , i.e., problem structures in this non-convex and coupled optimization problem can be leveraged to provide a very effective approxima- (OPT2) tion solution algorithm. Remark 3: The reference line transmit PSD is also static, and is set to the PSD adopted by the reference line in the absence Here the weight parameter , where means of self-crosstalk, and with a background noise of . This PSD user performs a pure selfish optimization, and means will be set based on the spectrum adaptation algorithm running on the modem when it operates in a fully selfish mode. In the the reference line’s rate will be maximized.5 In the synchronous case, it has been shown in [5] that the rate region of Problem simulations later in this paper, we use conventional single-user (OPT1) (in terms of and ) is convex in the asymptotic waterfilling to set , although in principle any static spectrum case with large number of tones. We can always find a value management algorithm could be used. Although this choice of of such that the optimal result of Problem (OPT2) is the reference noise and reference line transmit PSD is suboptimal, same as that of Problem (OPT1) (i.e., find a such that the it allows for an autonomous implementation, and as shown in solution of Problem (OPT2) satisfies ) as long Section VII, leads to a significant performance improvement as the latter is feasible. Thus the key challenge of the ASB over state-of-the-art autonomous algorithms, and in some sce- algorithm is to efficiently solve Problem (OPT2). The above narios leads to near-optimal performance. correspondence is not necessarily true in the asynchronous IV. ASB ALGORITHMS IN SYNCHRONOUS TRANSMISSION case. In that case, we can still use Problem (OPT2) as an approximation of Problem (OPT1) to derive an algorithm that In this section, we develop an ASB algorithm for the syn- achieves good performance. chronous case, where the achievable bit rates of user and the Remark 1: The crosstalk channels into the reference line reference line (from user ’s perspective) are given by (1) and and are modeled using the empirical models that have been (4), respectively. Since the transmissions on different tones are developed within the standards [12], [13], [16]. These are based orthogonal to each other here, we can use dual decomposition [18] to solve Problem (OPT2), defined for each user . Al- 5Problem (OPT2) can be derived from Problem (OPT1) using standard though Problem (OPT2) is nonconvex, we know from [5] that Lagrangian relaxation of user n’s target rate constraint, where the dual variable is chosen to be w a@I w A, which ranges from 0 to I. This the corresponding duality gap of Problem (OPT2) is zero in the weighted rate maximization representation was also used in [6] and [8]. asymptotic case where the total number of tones is large, thus 4246 IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 55, NO. 8, AUGUST 2007 solving the dual problem can lead to optimal primal solution. User then updates to enforce the total power constraint, We name the algorithm in this section as ASB-S1, where we and updates to enforce the target rate constraint. Both pa- solve Problem (OPT2) through a dual decomposition. Each user rameters can be found by a simple bisection search. Users then solves Problem (OPT2) by solving a nonconvex problem on iterate until all PSDs converge. The complete ASB-S1 algorithm each of the tones and choosing the dual variable (i.e., dy- is given in Algorithm I. namic price) such that the total power constraint is tight. Then users take turns to perform this optimization until the PSDs Algorithm 1: ASB Synchronous Model Version 1 (ASB-S1) converge. By incorporating the total power constraint into the objective function, we have the following relaxation of Problem (OPT2): 1: Initialize PSDs: , , . 2: repeat 3: for all user do 4: Initialize , 5: while do Here and needs to be chosen such that 6: . Then Problem (OPT2) can be solved by the fol- 7: Initialize , lowing unconstrained optimization problem: 8: while do 9: (6) 10: , . 11: if then 12: where denotes the PSD 13: else of all users except user . Further define 14: 15: end if (7) 16: end while 17: if then then it is clear that can be decomposed into a sum across 18: tones of , . As a result, Problem (6) can be 19: else decomposed into subproblems, one for each tone . The op- 20: timal PSD that maximizes is 21: end if 22: end while (8) 23: end for 24: until all users’ PSDs converge where . Although is Remark 4: The ASB algorithm leverages strong design nonconvex in , the maximization is over a scalar variable only, points from both OSB and IW. Like OSB, ASB uses a weighted and the optimal value can be easily found as follows. First rate-sum to account for the damage done to other lines within solve the first order condition, , which is equiva- the network when optimizing each line’s spectra. This weighted lent to rate-sum leads to a significant performance improvement over IW and in some scenarios leads to near-optimal performance. Like IW, ASB uses an iterative approach, optimizing the PSD of each user in turn. (9) Remark 5: The concept of a reference line has been employed extensively in heuristic-based DSM algorithms in the industry, including the reference PSD method that is currently Equation (9) can be simplified into a cubic equation which has mandated in the VDSL standards [12], [13], [16]. The reference three roots that can be written in close form. Then comparing PSD method is used in upstream VDSL transmissions to mit- the value of at each of these three roots, as well as checking igate the near–far problem. A similar technique has also been the boundary solutions and , we can find out recommended for downstream transmissions in order to pro- the corresponding value of .6 tect existing ADSL services from RT distributed VDSL [19]. There is a strong connection between ASB and the reference 6If an integer bitloading constraint is applied, then we can simply calculate PSD heuristics adopted in standards. Although the technique the PSD required to support each integer bitloading, and then evaluate the objective function v at the PSD corresponding to each integer bitloading value. for optimizing the PSD in ASB is different to that in the ref- The optimal choice is then selected. This allows integer bitloading constraints erence PSD method, both algorithms use representative “refer- to be incorporated without increasing complexity. Furthermore, spectral mask ence line” that shows what a typical line in the network looks constraints can also be applied in a straightforward fashion by disregarding any solution to the cubic equation that lies above the spectral mask, and adding the like, and how it should be expected to behave. In this paper, we spectral mask level itself to the set of points evaluated in the optimization. also develop proofs for convergence properties of ASB. CENDRILLON et al.: AUTONOMOUS SPECTRUM BALANCING 4247

Remark 6: In considering only a single reference line, the incremental amount of power a user can change on a tone at ASB algorithm makes an implicit assumption that, by protecting a time. In other words, represents the granularity of the the typical victim line in the binder, a user will indirectly pro- power shuffle, which trades off performance and convergence tect other shorter lines (i.e., stronger lines). The ASB algorithm speed. could be extended in a straightforward way to include multiple For each user with fixed , each search iteration consists reference lines, which does not impact the convergence proper- of two phases: subtraction phase and addition phase. In the ties and only leads to a small increase in complexity. For each subtraction phase, user reduces its PSD by on the tone extra reference line introduced into ASB, an extra local maxima that yields the maximum increase in (or the smallest will appear in the optimization of (8). ASB algorithm evalu- decrease if decreasing on any tone leads to a decreased ates the objective function at each local maxima and chooses objective). In the addition phase, user increases its PSD by the global maximum. As the frequency increases, we observe on the tone that yields the maximum increase in that the global optimal solution chosen by the ASB algorithm (or smallest decrease similar as in the subtraction phase). This jumps from a lower local optimal solution to a higher one. This iteration repeats until the net change of in the last is because, as frequency increases, the longest reference lines iteration (i.e., the sum of changes in both phases) is zero. becomes inactive due to weak direct channel in the high fre- Note that the net change of objective function will never be quency band, thus it is no longer necessary to protect this line. negative in a single iteration, since in the addition phase a A higher PSD is then chosen that corresponds to a higher local user can always add back to the same tone chosen in the optima. This new PSD will protect the second longest reference subtraction phase and recover the PSD level as in the previous line, which is now the weakest line in the system for that par- iteration. ticular frequency. When there are reference lines, the ASB The complete ASB-A1 algorithm is given in Algorithm 2. objective function exhibits up to local maxima. The first Line 7 computes user ’s PSD similar as in the synchronous local maxima correspond to protecting each of the reference case, given fixed transmission PSDs of other users, . Lines lines, while the st local maxima corresponds to the com- 8to10refine the value of several times by taking ICI into pletely selfish waterfilling solution, which is employed in the explicit consideration. For each value of granularity ,we very highest frequencies when all reference lines have switched apply the power shuffle (PS) subroutine (Algorithm 3) to up- off due to weak direct channels. To simplify presentation, in this date until convergence is reached, which occurs once no fur- paper we only focus on the approach of using a single reference ther greedy power swap can increase the objective. In a similar line. fashion to the barrier method [20], we use the optimal solution from the previous refinement as the initial position in the current refinement. By using diminishing values of , we achieve V. ASB ALGORITHMS IN ASYNCHRONOUS TRANSMISSION a much faster convergence rate and higher accuracy than can be In this section, we propose an ASB algorithm for the asyn- achieved with a single PSD granularity. Finally, user updates chronous case, where the achievable bit rates of user and in lines 11 to 15 using bisection search to make the target the reference line (from user ’s perspective) are given by rate constraint tight. (2) and (5). In this case, Problem (OPT2) is still non-convex and highly coupled due to crosstalk. Different from the syn- Algorithm 2: ASB Asynchronous Model Version 1 (ASB-A1) chronous case, a dual-based decomposition is not even applicable here since the PSD across different tones are coupled due to ICI. 1: Initialize PSDs: , , . We will introduce a greedy power shuffle algorithm into the 2: repeat ASB framework, where each user first initializes the PSD level 3: for all user do by solving Problem (OPT2) assuming synchronous transmis- 4: Initialize , sion (i.e., temporarily ignoring the ICI), then shuffle its PSD 5: while do (i.e., subtract a small amount from one tone and add it back 6: to another tone) to reach a locally optimal solution of Problem 7: Compute as Lines 7 to 16 in ASB-S1 (OPT2). Each user takes turns to perform this power shuffling 8: for all do until the PSDs converge. 9: . Let’s denote the objective function of Problem (OPT2) as 10: end for 11: if then 12: 13: else 14: 15: end if For notational simplicity, in the above expression we ignore 16: end while the dependence of on (which is assumed to be 17: end for fixed during user ’s PSD optimization). Define as the 18: until all users’ PSDs converge 4248 IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 55, NO. 8, AUGUST 2007

users fix their weights and aim at maximizing their rates under Algorithm 3: Power Shuffle (PS) subroutine a total power constraint [14].7 We notice that previous DSL lit- erature (e.g., [3], [5]–[8], [10], [11]) also focus on the RA mode 1: procedure when discussing convergence. Even when adapts to enforce 2: repeat target rate constraints, extensive simulations show that the algo- 3: . rithms proposed in this paper still converge. 4: for all do We first discuss the convergence of ASB-S1 in a two-user 5: case. The convergence of ASB-A1 has been briefly mentioned 6: in Proposition 1 for PS subroutine. We then consider the high 7: signal-to-noise ratio (SNR) regime for the reference line, under 8: end for which we prove stronger convergence results in both the syn- 9: chronous and asynchronous cases. 10: A. Convergence of ASB-S1 Algorithm 11: for all do 12: Here we discuss the convergence of the ASB-S1 algorithm, 13: where the nonconvexity of (9) makes it difficult to prove con- 14: vergence. In the two-user case, we can still show the following. Theorem 1: 15: end for Consider a two-user system with fixed and . There exists at least one fixed point of ASB-S1, and the algo- 16: rithm converges if users start from initial PSD values 17: or on all tones. 18: The proof of Theorem 1 uses supermodular game theory [21] 19: until and strategy transformation similar to [22], and is omitted here 20: return due to space limitation. Supermodular game theory can be used 21: end procedure to deal effectively with nonconvexity problems, and the convergence result in Theorem 1 does not require any condition on the The PS subroutine is specified in Algorithm 3. Line 3 finds the crosstalk channels. However, it is only for the case of fixed , set of tones on which a decrease of PSD will not lead to a nega- and users have to initialize their PSD at particular values. tive PSD. Lines 4 to 10 perform the subtraction phase, and lines 11 to 17 perform the addition phase. If a spectral mask constraint B. Convergence Under High SNR Regime of the is applied, then in the addition phase we exclude from consider- Reference Line ation any tones where addition would result in a spectral mask To reduce the computation complexity and gain more insight violation. Each user always achieves a better objective into the solution structure, we simplify the problem under high at the end of the PS subroutine, compared with the one achieved SNR approximation of the reference line as shown below. by using ASB-S1 algorithm before the PS subroutine. This is 1) Synchronous Transmission Case: The data rate of the ref- due to the monotonic increase of during the iterations erence line can be written as a linear function of the transmis- of the subroutine. Therefore, it is clear that the following is true. sion power of user under additional assumptions. First, from Proposition 1: The PS subroutine always converges. (4) we know that the reference line’s rate is a decreasing and The convergence of the ASB-A1 algorithm is difficult to show concave function in user ’s transmission power , and we can in general, due to the nonconvexity of Problem (OPT2) and the approximate with the following linear lower bound: fact that the PS subroutine can only reach a local optimal solution. In our simulations, however, the ASB-A1 algorithm always converges. Note also that, at the end of each iteration of the PS subroutine, the power constraint of user is always tight. This is because (10) we take away from one tone in the subtraction phase, and put it back into one tone in the addition phase. Thus the resource In other words, this gives the upperbound on the rate loss of the is always fully utilized and no power violation occurs. This is reference line due to the interference from user . Second, if we different from the bit-addition and bit-subtraction algorithms assume that the reference line operates in the high SNR regime in [10], where the power constraints are either loose or violated whenever it is active, i.e., if then , then (10) can during the whole process of the algorithm before convergence. be further simplified as

(11) VI. CONVERGENCE ANALYSIS 7The other categories of the spectrum balancing operation include Fixed Margin (FM) mode and Margin Adaptive (MA) mode, In FM, users try to In this section we prove convergence for various versions of minimize their power consumption under a target rate constraint. In MA, the ASB. We will only consider the rate adaptive (RA) mode, where users maximize their margins after achieving the target data rate. CENDRILLON et al.: AUTONOMOUS SPECTRUM BALANCING 4249 where is the indicator function and equals to one when Then, Problem (OPT2) becomes not only convex but also with event is true. Under (11), Problem (OPT2) becomes a convex a objective function that is separable across tones., i.e., optimization problem. In particular, user ’s maximization objective function on tone in (7) is approximated by

and the corresponding optimal PSD that solves Problem (OPT2) is given as thus the corresponding optimal PSD can be found in close form as

(13) (12) where is chosen to make the total power constraint tight . This is a generalization of the frequency selective where . This is a water-filling type of solu- waterfilling solution of ASB-S2. The complete ASB-A2 algo- tion, with different water-filling levels for different tones. We rithm is given in Algorithm 5. name it frequency selective waterfilling. Solution (12) is intuitively satisfying. The PSD for user should be smaller when the power constraint is tighter (i.e., is larger), or the crosstalk Algorithm 5: ASB-A2: ASB-A1 under high SNR regime channel to the reference line is higher, or the noise level on the reference line is smaller, or there is more interference 1: Replace Line 10 in Algorithm 1 with plus noise on the current tone. This leads to a second version of the ASB algorithm in the synchronous case, ASB-S2 algorithm as shown in Algorithm 4.

Algorithm 4: ASB-S2: ASB-S1 under high SNR regime

1: Replace Line 10 in Algorithm 1 with

3) Convergence of Algorithms ASB-S2/A2: We first consider the convergence in the two-user case where users sequentially optimize their PSD levels. Theorem 2: The ASB-A2 algorithm globally converges to the unique fixed point in a two-user system under fixed ,if . Proof of Theorem 2 is given in Appendix A. The key idea be- The ASB-S2 algorithm turns out to be a special case of the hind the proof is that the ASB-A2 algorithm leads to a contrac- ASB-A2 introduced next for the asynchronous case, of which tion mapping in the PSD updates, when the maximum product the convergence results will be presented in Section VI-B-3. of the crosstalk channel gains is small enough. One extreme 2) Asynchronous Transmission Case: Due to the coupling case is in a practical CO/RT mixed deployment case, where the induced by ICI, it is very difficult to find the global optimal so- crosstalk from CO to RT is negligible (i.e., lution of Problem (OPT2) in the asynchronous case. However, . if we also assume the high SINR regime and a linear approxi- Corollary 1: The ASB-S2 algorithm globally and geometri- mation of the bit per tone formula on the reference line as in the cally converges to the unique fixed point in a two-user system synchronous case, we have under fixed ,if . Corollary 1 recovers the convergence results for iterative water-filling in the two-user case [3] as a special case (by deactivating the reference line). We further extend the convergence results to a system with an arbitrary number of users. We consider both sequential 4250 IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 55, NO. 8, AUGUST 2007 and parallel PSD updates of the users. In the more realistic but harder-to-analyze parallel updates, time is divided into slots, and each user updates its PSD simultaneously with other users in each time slot according to (13) based on the PSDs from the previous time slot, and the is adjusted such that the power constraint is tight. Theorem 3: Assume , then the ASB-A2 algorithm globally and geometrically converges to the unique fixed point in an -user system under fixed , with either sequential or parallel updates. Fig. 2. Physical parameters of the DSL network. Proof of Theorem 3 is given in Appendix B. For ASB-S2 algorithm, we have the following. Corollary 2: If , then the come into close contact in the same binder and electromagnetic ASB-S2 algorithm globally and geometrically converges to the coupling can occur, is the frequency on tone (in megahertz), unique fixed point in an -user system under fixed , with ei- and is the distance from the transmitter of to the re- ther sequential or parallel updates. ceiver of line (in kilometers). A graphic illustration of the no- Corollary 2 recovers the convergence results for iterative tations is shown in Fig. 2. water-filling in an -user case with sequential updates (proved The convergence conditions for ASB-S2 is based on in [11]) as a special case. Interestingly, the convergence proof normalized channel gains . First con- for the parallel updates turns out to be simpler than that for sider the two user downstream ADSL case. For the channel sequential updates. Although convergence proofs are given for from the CO TX to the RT RX, , the case with a time-invariant update order, in our experience where is the length from the CO TX to the RT the algorithm always converges even if the modems are updated TX, and is the length of the RT line. In this case, we in a random fashion. Because of this, no centralized coordina- have tion is necessary for the updating of the individual modems, and . For ADSL, the maximum de- this can instead be triggered locally when a modem notices a ployment length is typically 5 km, so we can use this change in its local conditions, e.g., an increase in the measured to bound , i.e., noise spectrum as a new modem comes online. . For any particular value of , 4) Physical Meaning of Convergence Conditions: The con- the upperbound of can be maximized across , which is vergence conditions in Theorems 2 and 3 and Corollaries 1 and typically achieved at which corresponds to the highest 2 can be translated into constraints on the DSL network topolo- frequency at 1.1 MHz (i.e., interference is most severe on high gies. In downstream ADSL, the constraint can be translated into frequencies). Next, consider the channel from the RT into the the maximum distance between the transmitters of RT and the CO, CO, which limits the degree of crosstalk the RT transmitter can , where generate to CO receiver. In upstream VDSL, this means that . We can again maximize across (up to 1.1 lines cannot have lengths that are too different from one another, MHz) for any particular value of . To satisfy the conver- otherwise the near–far effect from the short lines into the long gence condition in Corollary 1, we need to find such that lines will cause severe crosstalk. in the synchronous To make the physical meaning more concrete, we consider a case. Using an SNR-gap of 12 dB, which includes a coding detailed DSL channel model that relates the channel gain to the gain of 4.2 dB and a noise margin of 6 dB, it turns out that all network topology. The magnitude of the direct channel can be values of satisfy the convergence conditions as modeled as , where is the line propagation con- shown in Fig. 3, which means ASB-S2 always converges in the stant, which depends on tone index , and is the line length. 2-user case for all deployment scenarios. In the user case, The value of is well understood, and very accurate models we find that, in the sufficient condition for convergence, the exist based on frequency, and the line diameter, construction, constraint on the maximum distance between the CO TX and materials, etc. The crosstalk channel, on the other hand, is not RT TX is too loose to be useful in practice. In our experience we as well understood.8 However, worst 1% case models for the find that ASB converges for a broad range of typical scenarios crosstalk channel have been developed, with which we can de- seen in DSL deployments. velop bounds that will guarantee convergence in 99% of lines. To be specific, the channel gain from transmitter to receiver VII. SIMULATION RESULTS in the worst 1% case crosstalk model is ([12], [13]) In this section, we show the performance of the ASB algo- . The constant , rithms, using a realistic simulator based on empirical channel is the length (in kilometers) over which line and models developed in standards and used extensively in the in- 8Significant progress has been made in developing more advanced crosstalk dustry [12], [13], [16]. In these simulations we use continuous channel models that take interpair distance and twisting imperfections into ac- bitloading and do not apply a bitcap. This is done since it results count [17], however this work requires detailed knowledge of the twisted pair geometry in order to estimate the crosstalk channels, something that is not avail- in PSDs that allow a more intuitive interpretation. It is also pos- able in an autonomous algorithm. sible to apply integer bitloading constraints, a bitcap, and PSD CENDRILLON et al.: AUTONOMOUS SPECTRUM BALANCING 4251

Fig. 5. Rate regions obtained by various DSM algorithms.

Fig. 3. Convergence conditions always satisfied in the two-user case since is chosen according to single-user waterfilling without consid- mx mx ` PXPdf. ering the interference from other users. Based on this reference line, we get the rate regions9 shown in Fig. 5. Observe that ASB achieves a near-optimal performance, almost identical to rate regions attained by the globally optimal OSB and ISB, and significant gains over IW. As a typical example, with a target rate of 1 Mb/s on user 1, the rate on user 4 reaches 7.3 Mb/s under ASB algorithm, which is a 143% increase (more than double) compared with the 3 Mb/s achieved by IW. Compared with IW, ASB exploits the special structure of the DSL channel and thus achieves much better performance. Since the direct channel gets worse with increasing frequency and Fig. 4. A four-user mixed CO/RT deployment topology (CP denotes customer length, long lines cannot effectively utilize high frequencies. premises). The crosstalk channel strength, on the other hand, increases with frequency. In the ASB algorithm, the RT lines transmit with high power in the low frequencies where there is little crosstalk, masks to ASB with negligible increase in complexity. We only reduce power in the middle frequencies to protect the reference simulate the performances of the ASB-S1 and ASB-A1 algo- line, and switch to high power again in the high frequencies rithms, which do not involve any high SNR assumptions. where reference line is not active. In the IW algorithm, however, the power allocation is as follows: A. Synchronous Transmission Case Here we summarize a typical numerical example, representative of a large set of experiments we conducted, comparing the performance of the ASB-S1 algorithms with IW, OSB, and ISB in the synchronous transmission case. A four-user mixed where the adjustable part is the same on all fre- CO/RT scenario has been selected to make a comparison with quencies. User first adjusts such that its total power con- the highly complex OSB algorithm possible. As depicted in straint is tight. If the achieved rate is larger than the target Fig. 4, user 1 is CO line, while the other three users are RT lines. rate , it performs equal power-backoff at all frequen- ANSI noise model A [23] has been used, which consists of 16 cies (i.e., increase the value of ), which unnecessarily reduce ISDN, 4 HDSL and 10 conventional (non-DSM capable) ADSL the power at the very low (where little crosstalk is generated to disturbers. the CO line) and high frequencies (where the CO line is inac- Due to the different distances among the corresponding trans- tive). As a result, the IW algorithm leads to highly suboptimal mitters and receivers, the RT lines generate strong interference performance, especially in near–far scenarios. As an example, into the CO line, while experiencing very little crosstalk from we plot the PSD allocations under the ASB, IW and ISB/OSB the CO line. The target rates of users 2 and 3 have both been set algorithms in Fig. 6, with the achievable rates of four users as to 2 Mb/s. User 4 changes its target rate from 0 to 8 Mb/s, and , , for IW and user 1 (the CO line) does not have a target rate constraint and 7.3 Mb/s for ASB-A1/ISB/OSB. always sets its weight coefficient equal to unity in ASB-S1 We also summarize a typical simulation of the ASB and IW (i.e., maximizes its own rate without protecting the reference algorithms in a network with 10 lines, with the line length equal line). The reference line is chosen to match the longest line in to 5 km for the CO line, and 2 km to 4.5 km for the RTs in the network in terms of background noise and crosstalk channel gains with users in the network. The reference line transmit PSD 9Note that only ASB uses the reference line idea. 4252 IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 55, NO. 8, AUGUST 2007

Fig. 7. Mixed CO/RT distribution.

speciﬁed as 1.6 Mb/s. With this in mind, the target rates for the RT modems, which are set equally on all RTs, are reduced until the CO modem achieves its target rate. With IW, the RTs are forced to reduce their rates to 0.8 Mb/s in order for the CO to achieve it’s target. With ASB, due to the more intelligent allocation of the RT transmit spectra, the RTs can maintain a rate of 2.0 Mb/s while still ensuring that the CO modem achieve 1.6 Mb/s. The ASB algorithm achieves a gain of 122% in the RT rate with respect to IW.

B. Asynchronous Transmission Case Now consider the case of asynchronous transmission. Here we summarize a typical numerical example comparing the performances of the ASB-A1 and ASB-S1 algorithms. As depicted in Fig. 7, the scenario consists of downstream transmission with two ADSL modems, one 5 km CO line, and one 3 km RT line. The RT TX is deployed 4 km downstream from the CO TX. Figs. 8 and 9 show an example of the PSDs generated by ASB-A1 and ASB-S1. The target rate for the RT is set to 3.85 Mb/s. Using ASB-S1, which does not take the effects of the ICI into account when optimizing the transmit spectra, the CO achieves 1.3 Mb/s. Using ASB-A1, the CO rate increases to 1.6Mb/s.With ASB-A1,thetransmitpowerisshiftedfurtherinto thehigh-frequenciestopreventexcessiveICItotheCOline.Also, since the ICI creates an unavoidable “noise”floor of at around 90 dBm/Hz, it is possible to increase the transmit PSD between 340 KHz and 680 KHz with minimal degradation to the CO line. Fig. 10 shows the increase in performance relative to IW achieved by ASB-S1 and ASB-A1, respectively, in an asynchronous environment. As we see, even when the modems are not synchronized, ASB-S1 achieves significant gains over IW. Furthermore, if the transmit spectra are further refined through the application of ASB-A1, even further performance gains are possible. For example, if the CO rate is set at 1.4 Mb/s, applying ASB-S1 increases the RT rate by 48% over IW. Applying ASB-A1 leads to a further increase in the RT rate of 186%, leading to a total gain of 234% over IW.

Fig. 6. Transmit spectra with synchronous transmission. (a) ASB-S1; (b) IW; C. Sensitivity Analysis of the Reference Line Choices (c) ISB/OSB. In previous simulation examples, we choose the reference steps of 0.3125 km. The RTs are correspondingly located 2 km line to match the longest line in the binder. Here we study the to 4 km from the CO. The target rate for the CO modem was robustness of the performance to the choice of reference line CENDRILLON et al.: AUTONOMOUS SPECTRUM BALANCING 4253

Fig. 8. Transmit spectra with asynchronous transmission: ASB-S1. Fig. 11. Sensitivity of ASB-S1 to choice of reference line length.

Fig. 11 shows the achievable rate regions with the different reference line length. Obviously, optimal performance is achieved by setting the reference length to 5000 m, the length of the weaker CO distributed line. We notice that the performance is relatively insensitive to the choice of the reference line length, especially during a broad range of 4050 m to 6000 m. Only when the reference line becomes extremely inaccurate (i.e., around 4020 m or less), which seldom happens in practice, performance starts to degrade rapidly. This is because with a 4020 m reference line, the ASB algorithm assumes that the RT TX is located only 20 m from the reference line RX (recall that the RT RX is actually 4000 m from the CO RX). This will lead to a huge crosstalk channel from the RT to the reference line, and the RT is forced to reduce power in the entire frequency band within which the CO transmits. As can be seen, the performance of ASB is quite insensitive to the Fig. 9. Transmit spectra with asynchronous transmission: ASB-A1. mismatch between the length of the reference line and the length of the longest line in the binder. And even for quite big errors in reference line settings, the attainable rate region by ASB is still much larger than IW. Mathematically, this means that the dependence of the values of the local maxima of this nonconvex optimization problem on the crosstalk channel coefﬁcients is sufﬁciently insensitive for the observed robustness to hold.

VIII. COMPLEXITY ANALYSIS Here we compare the complexity of ASB-S1 algorithm with the IW algorithm, which is summarized in Table II.10 Running time is measured based on the results of Matlab programs running on an MS-windows machine with a P4-2.8 GHz processor. Real time operations based on hardware implementation would be several orders of magnitude faster. The example we simu- lated includes a total of tones and lines. Cy- Fig. 10. Performance gains of ASB-S1 and ASB-A1 over IW under asyn- cles till convergence is number of outer-cycles required through chronous transmission. all of the users before convergence occurs. We typically see that only three outer-cycles are necessary for the rates to converge within 1% of the previous cycle. length. We run simulations in a two user scenario as in Fig. 7, assuming that the modems operate synchronously. For these sim- 10The complexity result of ASB-A1 algorithm was summarized in Table I, and the corresponding analysis details are omitted due to space limitation. The ulations, we vary the length of the reference line, but hold the complexities of OSB, ISB, and SCALE are too high to be comparable to ASB length of the CO line in the binder at 5 km. or IW, and are omitted here. 4254 IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 55, NO. 8, AUGUST 2007

TABLE II plexity order and centralized schemes of OSB and ISB, which ALGORITHM COMPLEXITY do not offer much rate region gains over ASB.

IX. CONCLUSION This paper developed a suite of DSM algorithms referred to as ASB, which are autonomous, have a low complexity and achieves significant performance gains over the prior state-of- the-art autonomous algorithms such as Iterative Waterfilling. In typical scenarios ASB also achieves near-optimal performance, A. Complexity Analysis for IW which was previously only possible with the centralized, and Iterative waterfilling consists of an outer cycle that iterates highly complex Optimal Spectrum Balancing algorithm. through users, and an inner loop that adjusts the total power of The convergence of the ASB is proven for an arbitrary the current user until the target data rate is achieved. For each number of users in rate-adaptive mode. In particular, ASB user , we use a bisection on within the inner loop, which includes IW as a special case, thus the convergence proof of is both efficient and robust. In the inner loop, each user needs our algorithm extends and generalizes the convergence proof to find the power required to hit its target rate constraint. Typi- of IW. ASB can improve system performance with both syn- cally achieving a precision of in the total power setting is chronous and asynchronous transmission, where the latter is a sufficient to hit the target rate with high accuracy. This requires particularly under-explored research area where only limited, iterations of bisection search. high-complexity heuristics were previously available. For each iteration within the inner loop under a fixed value of The key concept that enables ASB to successfully tackle , a standard waterfilling algorithm must be applied with the the non-convex, coupled, and high-dimensional optimization following complexity11: problem is the reference line, which allows each user to opti- 1) Find the optimal water level such that the total power con- mize its transmit spectra independently. Each user attempts to straint is satisfied and allocated power is positive on all ac- achieve its own target rate whilst minimizing the degradation tive tones: operations [24]. caused to the reference line, which represents a typical victim 2) Calculate based on the optimal water level: line within the DSL network. ASB applies this approach of operations. “static pricing” coordination in a rigorous manner with prov- 3) Calculate the corresponding bitloading: operations. able theoretical properties, leading to a significantly enlarged Hence the total complexity of a single waterfilling is oper- rate region compared with IW. The reference line idea can be ations, where one operation is either an addition or a multipli- readily implemented using the reference lines already devel- cation. Considering the 34 iterations of the bisection search, the oped within standards. Although we have focused primarily on iteration through all of the users, and the iteration of the whole ADSL in this paper, ASB is also applicable in VDSL systems. process until convergence, the total complexity of IW is then: , where is the number of cycles APPENDIX required until convergence. A. Proof of Theorem 2 B. Complexity Analysis for ASB-S1 The following Lemma is useful for proving Theorem 2. ASB-S1 consists of three levels of iterations, with the out- Lemma 1: Consider any non-decreasing function and ermost cycle iterating through users. Within each cycle, each non-increasing function . If there exists a unique such users runs an outer loop where it updates until the target data that , and the functions and are strictly rate is achieved, and an inner loop where it updates until the increasing and strictly decreasing at respectively, then total power constraint is satisfied. The bisection search is used . in both loops. To achieve a precision of in both and Proof of Lemma 1: For any , , we need a total of iterations. Within each it- . Similarly for any , eration, the complexity is dominated by finding the roots of a . It then can be cubic equation [e.g., solving (9)], which requires 44 operations verified that . in total [25]. This has to be repeated on all tones, leading to a Denote as the PSD of user on tone after iteration , total complexity of . Hence the total complexity of ASB-S1 where is satisfied at the end of any iteration for is . High SNR approximation any user . One iteration is defined as one round of updates of would further reduce the operations count. all users. The PSD update in the two-user case can be written as It is important to realize that the order of complexity for ASB follows: is the same as IW: , and the actual running time of ASB is still well within the bounds for practical implementation. This implementation viability is in sharp contrast to the higher com-

11Also, the inverse Channel-Signal-to-Noise-Ratio (CSNR) must be calculated, and the tones sorted according to the CSNR. However this only needs to be done once for each outer cycle, and can be re-used for all inner-loop itera- (14) tions. Hence this has a minimal impact on complexity. CENDRILLON et al.: AUTONOMOUS SPECTRUM BALANCING 4255 where , , and , and . Also define . Without loss of generality, we assume that the total power constraint is always satisfied at the end of any iteration. In general, the total power constraint needs not to be tight, e.g., (19) when summation of (which is determined by (12)) over all tone is less than the power constraint even when . This might happen in the case where is small enough (i.e., user ’s target rate is small). However, we can make the power constraint tight in this case by defining an extra “virtual tone”. The data rate achieved by user on the virtual tone is , where is a very small number and is the PSD allocated (20) to the virtual tone. Furthermore, the reference line is chosen to be inactive on the virtual tone (i.e., ). Now from the perspective of any actual line, loading power on the virtual tone has very small yet positive impact on its own total rate (with very small value ), and has no impact on the reference line’s rate. Hence the user will always take any left over power and load (21) onto the virtual tone, and always operate at full power. Then it is clear that

(15)

Also deﬁne (22)

(23)

where (16) follows from the definition of and , (17) follows by using Lemma 1 and letting , (18) follows from the definition of and , the expression of in (14), and and the fact that and for any and , (19) follows by exchanging indexes and , (20) follows by using for all and , (21) follows by using the circulant property of , i.e., , (22) by applying the argu- ments from (16) to (21) again, and finally (23) follows by the condition in Theorem 2. This shows that the ASB-A2 algorithm is a contraction mapping form an initial PSD values. It can be shown that is a norm, thus ASB-A2 where , and . It is clear that globally converges to a unique fixed point [26, p. 183]. ( , respectively) is non-decreasing (non-increasing) in , and strictly increasing (strictly decreasing) B. Proof of Theorem 3 at (unless , We first prove the convergence in the parallel update case. which means the PSD converges). From (15) we always have The PSD of user in tone after iteration is . Now we can show that

(16) (17)

The rest of the proof can be obtained similar as in Theorem 2 with the following: (see the equation at the top of the next page). For the sequential update case, the convergence can be proved (18) by combining Lemma 1 and proof of Theorem 3.4.1 in [11]. First, deﬁne 4256 IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 55, NO. 8, AUGUST 2007

, and . Using induction, we [14] T. Starr, J. Cioffi, and P. Silverman, Understanding Digital Subscriber can find an matrix such that . Line Technology. Englewood Cliffs, NJ: Prentice-Hall, 1999. [15] F. Sjoberg, M. Isaksson, R. Nilsson, and P. Borjesson, “Zipper: A du- The final step is to show the maximum eigenvalue of matrix plex method for VDSL based on DMT,” IEEE Trans. Commun., vol. is less than 1, which guarantees that ASB-A2 algorithm is 47, no. 8, p. 1245, 1999. an contraction mapping in the sequential updates. Details are [16] Very High Speed Digital Subscriber Line Transceivers 2, ITU Draft Std. G.993.2, 2006. omitted due to space limitations. [17] B. Lee, “Binder MIMO channels,” Ph.D. dissertation, Stanford Univ., Stanford, CA, Nov. 2004. ACKNOWLEDGMENT [18] S. Boyd and L. Vanderberghe, Convex Optimization. Cambridge, MA: Cambridge Univ. Press, 2004. The authors would like to acknowledge helpful discussions [19] Spectrum Management for Loop Transmission Systems, Issue 2, ANSI from J. Cioffi, A. Fraser, A. H. Mohsenian-Rad, D. Palomar, Std. T1.417, 2003. C. W. Tan, W. Yu, and J. (Steve) Yuan. [20] D. Bertsekas, Nonlinear Programming, 2nd ed. Belmont, MA: Athena Scientific, 1999. [21] D. M. Topkis, Supermodularity and Complementarity. Princeton, NJ: REFERENCES Princeton Univ. Press, 1998. [1] R. Cendrillon, G. Ginis, and M. Moonen, “Improved linear crosstalk [22] J. Huang, R. Berry, and M. L. Honig, “Distributed interference com- precompensation for downstream VDSL,” in Proc. IEEE Int. Conf. pensation in wireless networks,” IEEE J. Sel. Areas Commun., vol. 24, Acoust., Speed Signal Process. (ICASSP), 2004, pp. 1053–1056. no. 5, pp. 1074–1084, May 2006. [2] G. Ginis and J. M. Cioffi, “Vectored transmission for digital sub- [23] Noise Models for VDSL Performance Verification, ANSI-77E7.4/99. scriber line systems,” IEEE J. Sel. Areas Commun., vol. 20, no. 5, pp. 438R2, Dec. 1999. 1085–1104, 2002. [24] J. Cioffi, Advanced Digital Communication EE379C Course [3] W. Yu, G. Ginis, and J. Cioffi, “Distributed multiuser power control for Reader, Chapter 4—Multi-Channel Modulation [Online]. Avail- digital subscriber lines,” IEEE J. Sel. Areas Commun., vol. 20, no. 5, able: http://www.stanford.edu/class/ee379c/readerfiles/chap4.pdf pp. 1105–1115, Jun. 2002. [25] R. Nickalls, “A new approach to solving the cubic: Cardan’s solution [4] Z.-Q. Luo and J.-S. Pang, “Analysis of iterative waterfilling algorithm revealed,” Mathematical Gazette, vol. 77, pp. 354–359, 1993. for multiuser power control in digital subscriber lines,” EURASIP J. [26] D. P. Bertsekas and J. N. Tsitsiklis, Parallel and Distributed Com- Appl. Signal Process., 2006, vol. 2006, Article ID 24012, pp. 1"C10. putation: Numerical Methods. Englewood Cliffs, NJ: Prentice-Hall, [5] R. Cendrillon, W. Yu, M. Moonen, J. Verlinden, and T. Bostoen, “Op- 1989. timal multiuser spectrum balancing for digital subscriber lines,” IEEE Trans. Commun., vol. 54, no. 5, pp. 922–933, May 2006. [6] R. Cendrillon and M. Moonen, “Iterative spectrum balancing for digital subscriber lines,” in IEEE ICC, 2005. [7] R. Lui and W. Yu, “Low-complexity near-optimal spectrum balancing Raphael Cendrillon (S’02–M’04) was born in for digital subscriber lines,” in IEEE ICC, 2005. Melbourne, Australia, in 1978. He received the [8] J. Papandriopoulos and J. Evans, “Low-complexity distributed algo- Electrical Engineering (First Class Hons.) degree rithms for spectrum balancing in multi-user DSL networks,” in IEEE from the University of Queensland, Australia, in ICC, 2006. 1999 and the Ph.D. degree in electrical engineering [9] J. M. Cioffi, W. Rhee, M. Mohseni, and M. H. Brady, “Band preference (summa cum laude) from the Katholieke Universiteit in dynamic spectrum manag,” in EUSIPCO Conf. Signal Process.,Vi- Leuven (K.U. Leuven), Belgium, in 2004. enna, Austria, Sep. 2004. In 2002, he was a Visiting Scholar with the In- [10] V. M. K. Chan, “Multiuser detection and spectrum balancing for dig- formation Systems Laboratory, Stanford University, ital subscriber lines,” Master’s thesis, Dep. Elect. Comp. Eng., Univ. Stanford, CA, with Prof. J. Cioffi. In 2005, he was Toronto, Toronto, 2005. a Postdoctoral Research Fellow with the University [11] S. T. Chung, “Transmission schemes for frequency selective gaussian of Queensland, Australia. During this period, he was also a Visiting Research interference channels,” Ph.D. dissertation, Stanford Univ., Stanford, Fellow with Princeton University, Princeton, NJ, with Prof. M. Chiang. Cur- CA, 2003. rently, he is a Senior DSP Engineer with Marvell Hong Kong, Ltd. [12] Very High Speed Digital Subscriber Line (VDSL); Functional Require- Dr. Cendrillon was awarded the Alcatel Bell Scientific Prize in 2004, the ments, Rev. V.1.3.1, ETSI Std. TS 101 270-1, 2003. IEEE Travel Grants in 2003, 2004, and 2005, the K.U. Leuven Bursary for Ad- [13] Very-High Bit-Rate Digital Subscriber Lines (VDSL) Metallic Inter- vanced Foreign Scholars in 2004, and the UniQuest Trailblazer Prize for Com- face, ANSI Std. T1.424, 2004. mercialization in 2005. CENDRILLON et al.: AUTONOMOUS SPECTRUM BALANCING 4257

Jianwei Huang (S’00–M’06) received the B.S. de- Marc Moonen (M’94–SM’06–F’07) received the gree in electrical engineering from Southeast Univer- electrical engineering degree and the Ph.D. degree sity, Nanjing, China, in 2000 and the M.S. and Ph.D. in applied sciences from Katholieke Universiteit degrees in electrical and computer engineering from Leuven (K.U.Leuven), Belgium, in 1986 and 1990, Northwestern University, Evanston, IL, in 2003 and respectively. 2005, respectively. Since 2004, he has been a Full Professor at the He is a Postdoctoral Research Associate with the Electrical Engineering Department of Katholieke Department of Electrical Engineering, Princeton Universiteit Leuven, where he is currently heading University, Princeton, NJ. In 2004 and 2005, he a research team of 16 Ph.D. candidates and post- worked in the Network Advanced Technology docs, working in the area of numerical algorithms Group at Motorola, both as a full-time summer and signal processing for digital communications, intern and a part-time researcher. In 1999, he worked as a summer intern with wireless communications, DSL, and audio signal processing. the Department of Change Management, GKN Westland Aerospace Co., Ltd. Dr. Moonen received the 1994 K.U.Leuven Research Council Award, the His main research interests lie in the area of communications and networking, 1997 Alcatel Bell (Belgium) Award (with Piet Vandaele), and the 2004 Alcatel with speciﬁc areas including cognitive radio networks, wideband OFDM and Bell (Belgium) Award (with Raphael Cendrillon) and was a 1997 “Laureate CDMA systems, wireless medium access control, multimedia communications, of the Belgium Royal Academy of Science.” He received a journal Best Paper cooperative communications, and wired DSL broadband access networks. Award (with G. Leus) from the IEEE TRANSACTIONS ON SIGNAL PROCESSING Dr. Huang is an Associate Editor of the (Elsevier) Journal of Computer & and (with S. Doclo) from Elsevier Signal Processing. He was Chairman of the Electrical Engineering, the Lead Guest Editor of the “Game Theory in Com- IEEE Benelux Signal Processing Chapter from 1998 to 2002 and has been a munication Systems” special issue of the IEEE JOURNAL OF SELECTED AREAS European Association for Signal, Speech and Image Processing (EURASIP) IN COMMUNICATIONS, the Lead Guest Editor of the “Collaboration and Opti- AdCom Member since 2000 and a member of the IEEE Signal Processing So- mization in Multimedia Communications” special issue of the Journal of Ad- ciety Technical Committee on Signal Processing for Communications. He has vances in Multimedia, and a Guest Editor of the “Cross-Layer Optimized Wire- served as Editor-in-Chief for the EURASIP Journal on Applied Signal Pro- less Multimedia Communications” special issue of the Journal of Advances in cessing from 2003 to 2005 and has been a member of the editorial board of Multimedia. He is the recipient of a 2001 Walter P. Murphy Fellowship at North- IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS I from 2002 to 2003 and IEEE western University and a 1999 Chinese National Excellent Student Award. SIGNAL PROCESSING MAGAZINE from 2003 to 2005. He is currently a member of the editorial board of Integration, the VLSI Journal, EURASIP Journal on Ap- plied Signal Processing, EURASIP Journal on Wireless Communications and Networking, and Signal Processing. Mung Chiang (S’00–M’03) received the B.S. (Honors) degree in electrical engineering and mathematics, and the M.S. and Ph.D. degrees in electrical engineering from Stanford University, Stanford, CA, in 1999, 2000, and 2003, respectively. He is an Assistant Professor of Electrical En- gineering and an afﬁliated faculty member of the Program of Applied and Computational Mathematics and of the Computer Science Department, Princeton University, Princeton, NJ. He conducts research in the areas of optimization of communication systems, theoretical foundation of network architectures, algorithms in broadband access networks, and stochastic models of communications. Prof. Chiang has been awarded a Hertz Foundation Fellow, and received the Stanford University School of Engineering Terman Award, SBC Commu- nications New Technology Introduction Contribution Award, NSF CAREER Award, an ONR Young Investigator Award, and the Princeton University Howard B. Wentz Junior Faculty Award. His January 2005 IEEE JOURNAL ON SELECTED AREAS IN COMMUNICATIONS paper became the Fast Breaking Paper in Computer Science in 2006 according to ISI’s citation frequency. He coauthored the Best Student Paper at IEEE Globecom 2006. He is the Lead Guest Editor of the Special Issue of the IEEE JOURNAL ON SELECTED AREAS IN COMMUNICATIONS on “Nonlinear Optimization of Communication Systems,” a Guest Editor of the Joint Special Issue of the IEEE TRANSACTIONS ON INFORMATION THEORY and the IEEE/ACM TRANSACTIONS ON NETWORKING on “Networking and Information Theory,” an Editor of the IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, a Program Co-Chair of the 38th Conference on Information Sciences and Systems, and a coeditor of the new Springer book series Optimization and Control of Communication Systems.